Abstract
Let I be a 1-shift-invariant ideal on N with the Baire property. Assume that a series ∑nxn with terms in a real Banach space X is not unconditionally convergent. We show that the sets of I-convergent subseries and of I-convergent rearrangements of a given series are meager in the respective Polish spaces. A stronger result, dealing with I-bounded partial sums of a series, is obtained if X is finite-dimensional. We apply the main theorem to series of functions with the Baire property, from a Polish space to a separable Banach space over R, under the assumption that the ideal I is analytic or coanalytic.
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